let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * ln ) implies ( - (cosec * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) ) ) )
assume A1: Z c= dom (cosec * ln ) ; :: thesis: ( - (cosec * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) ) )
then A2: Z c= dom (- (cosec * ln )) by VALUED_1:8;
A3: cosec * ln is_differentiable_on Z by A1, FDIFF_9:15;
then A4: (- 1) (#) (cosec * ln ) is_differentiable_on Z by A2, FDIFF_1:28, A;
for x being Real st x in Z holds
((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) )
assume A5: x in Z ; :: thesis: ((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 ))
((- (cosec * ln )) `| Z) . x = ((- 1) (#) ((cosec * ln ) `| Z)) . x by A3, FDIFF_2:19, A
.= (- 1) * (((cosec * ln ) `| Z) . x) by VALUED_1:6
.= (- 1) * (- ((cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )))) by A1, A5, FDIFF_9:15
.= (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) ;
hence ((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) ; :: thesis: verum

end;

hence ( - (cosec * ln ) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (cosec * ln )) `| Z) . x = (cos . (ln . x)) / (x * ((sin . (ln . x)) ^2 )) ) ) by A4; :: thesis: verum